Karl Popper and Critical Rationalism

Karl Popper and Critical Rationalism

<!--<!--“Critical Rationalism” is the name Karl Popper (1902-1994) gave to a modest and self-critical rationalism. He contrasted this view with “uncritical or comprehensive rationalism,” the received justificationist view that only what can be proved by reason and/or experience should be accepted. Popper argued that comprehensive rationalism cannot explain how proof is possible and that it leads to inconsistencies. Critical rationalism today is the project of extending Popper’s approach to all areas of thought and action. In each field the central task of critical rationalism is to replace allegedly justificatory methods with critical ones.

Section 2 explains how critical rationalism arose out of the breakdown of Popper’s first justificationist attempt to account for scientific progress. Section 2 also presents Popper’s first application of his non-justificationist perspective to new fields in his The Open Society and Its Enemies. Section 3 first explains Joseph Agassi’s view of the critical appraisal of metaphysical theories in scientific contexts as well as his view of piecemeal rationality, and secondly portrays William Bartley’s more comprehensive view of non-justificationism. Section 4 discusses Imré Lakatos’s extension of critical rationalism to mathematics. Section 5 portrays Hans Albert’s systematic version of critical rationalism. His perspective incorporated results of Popper, Agassi and Bartley and extended them to social and political theory. Section 6 suggests that Mario Bunge’s fallibilism ─ which he developed independently of critical rationalists ─ is sufficiently close to their views to count here: he develops critical tools for achieving progress without justification in virtually all areas of thought. Section 7 discusses attempts to develop critical rationalism in new and simpler ways. These views seek to do without frameworks and methodological rules; their originators are Jagdish Hattiangadi, Gunnar Andersson, and David Miller. These theories deprive rational thought of needed steering mechanisms. Section 8 presents the reintroduction of forms of justification designed to be compatible with Popper’s criticism of induction. These have been developed by Alan Musgrave, Volker Gadenne and John Watkins. Section 9 explains how Popper’s emphasis on the importance of methodological rules in science has led to a critical rationalist sociology of science. The main task of this sociology of science is to examine existing rules and methods as furthering or hindering research. Section 10 calls attention to the alternative philosophical anthropology which Agassi has proposed as a framework for critical rationalism. Whereas Popper saw rationality as contrary to human nature’s craving for security, Agassi sees rationality as natural, but partial and improvable. Section 11 describes how Popper’s original political manifesto in The Open Society and Its Enemies has led to attempts to use his arguments to defend both right-leaning and left-leaning political theories. Section 12 returns to Popper’s early researches in educational theory. His philosophy led to concerted efforts to develop a new pedagogy which emphasizes active problem solving as the best learning method. This pedagogy should promote autonomy and critical thinking. Section 13 concludes with the suggestion that the success or the failure of the project of substituting critical for allegedly justificatory methods has still to be judged.

Table of Contents

1. Introduction

Critical rationalism emerged from research by the Würzburg school of psychology. This school sought to develop a deductivist philosophy of science to complement their deductivist psychology. While working on this program, Karl Popper stumbled onto a non-justificationist theory of scientific knowledge: he explained the growth of knowledge without proof. Non-justificationism, that is, the theory that no theory can be proven, is at least as old as Socrates, but Popper’s version of it is the first that also purports to explain the growth of knowledge. Popper and other critical rationalists took on the project of explaining the growth of knowledge without justification. This project has produced various competing theories of rationality and has been extended to many fields. This article will concentrate on the internal logic and problems involved in the development of critical methods capable of producing the growth of knowledge.

Of the numerous justificationist predecessors let only this be said. The overwhelming majority of those who comment on critical rationalism claim that critical rationalism is somehow incoherent and that inductivism is better. A major exception was Bertrand Russell. He appreciated the logical strength of critical rationalism and knew the logical weakness of induction. Nevertheless he clung to induction. He thought that critical rationalism was a philosophy of despair. Whether his judgment of critical rationalism was correct depends on whether its development can bring progress. To show this progress, new critical rationalist ideas are described and presented below. This should provide an answer to Russell that he amply deserves.

2. Popper and Non-Justificationism

Inductive inferences have observations as premises and theories as conclusions. They are notoriously invalid but often are deemed unavoidable. Critical rationalism views them as unnecessary. This point of view grew gradually out of Karl Popper’s attempt to describe science without their use in Die beiden Grundprobleme der Erkenntnistheorie (1932-33 ), where he still operated within the framework of justificationism, that is, while viewing the aim of scientific method as the proper (justified) assessment of the truth value of certain sentences. He hoped to build a theory of the proper assessment of sentences, that is, of the possibility of proving the truth or falsity of some sentences. He began with the fact that a theory is false if it contradicts a singular sentence describing some observation reports. Popper then said that such singular sentences were veridical, that is, truthful as opposed to illusory, so they may be used to produce final proofs of the falsity of some universal sentences. For example, the singular sentence, “That swan is black,” if it is a true report of some observation, can be used to produce a final proof of the falsity of the universal sentence, “All swans are white.” But, he argued, proof of universal sentences or the demonstration that they are probable requires inductive inferences. As a consequence no such putative proof can be valid.

Popper himself found the theory he presented in Die beiden Grundprobleme der Erkenntnistheorie without chapter 5 inadequate for three reasons. The first reason is that singular statements are not veridical. He began work on this problem in chapter 5 of Die beiden Grundprobleme. This chapter contains a theory of science which differs on important points from the theory found in the rest of that volume. The second reason that Popper’s first attempt broke down is that one can circumvent refutations by ad hoc stratagems, as Hans Reichenbach quickly pointed out in a note which responded to Popper’s first publication of his view in Erkenntnis. The third reason was Popper’s inability to handle the problem of the demarcation of science from non-science with his idea that we show how science properly assigns truth values to sentences with no inductive inference. On a justificationist theory of the task of the philosophy of science such as Reichenbach’s, which was identical to Popper’s theory as he wrote Die beiden Grundprobleme without chapter 5, science should be demarcated by the proper assignment of truth values: science is the set of sentences with justifiably assigned truth values. The task of the philosophy of science is to explain how these assignments are properly made. (Reichenbach said the calculus of probabilities serves that purpose.) Popper argued that it is not possible to properly assign either the truth value True or some degree of probability to universal sentences. He called such sentences “fictions”, which is a term he had earlier taken over from Hans Vaihinger. On the theory presented in Die beiden Grundprobleme without chapter 5, after science had done its job, there were still, on the one hand, some fictions which ought to be deemed scientific such as the theories of the Würzburg school in psychology and, perhaps, as he said later, Einstein’s physics, and, on the other hand, other fictions which should be deemed unscientific, such as the psychologies of Freud and Adler. He could not distinguish between these two sets of theories within his justificationist framework, since, on this view, only proofs or refutations of these theories could do that. He asserted, however, that no proof was possible and refutations could establish only the falsity of universal propositions.

As a consequence of these three difficulties Popper developed an entirely different theory of science in chapter 5, then in Logik der Forschung. In order to overcome the problems his first view faced, he adopted two central strategies. First, he reformulated the task of the philosophy of science. Rather than presenting scientific method as a tool for properly assigning truth values to sentences, he presented rules of scientific method as conducive to the growth of knowledge. Apparently he still held that only proven or refuted sentences could take truth values. But this view is incompatible with his new philosophy of science as it appears in his Logik der Forschung: there he had to presume that some non-refuted theories took truth values, that is, that they are true or false as the case may be, even though they have been neither proved nor refuted. It is the job of scientists to discover their falsity when they can. So, he worked around the difficulty posed by the fact that, on the one hand, he had to assume that theories were refutable and thus had truth values, whereas, on the other hand, he thought that only proven or refuted theories had truth values at all. He argued that his view could be interpreted as realist or as antirealist. He hedged his bets as best he could and appealed to Mach, who had stipulated that one should avoid participation in any metaphysical dispute.

In Logik der Forschung Popper solved his three initial difficulties in the following ways. First, instead of claiming that singular sentences were veridical, he said that basic statements are only provisionally accepted, provided that they were repeatable and so testable. He thereby introduced the following rule: consider only repeatable basic statements. He claimed that the provisional acceptance of basic statements does not disqualify them as refutations of theories—no longer simply universal sentences—because for the most part we can agree on which basic sentences we provisionally assume to be true. Second, he proposed the rule that one should always replace some theory which is contradicted by a basic statement by whichever new alternative has the highest degree of falsifiability. This rule should guarantee that refutations lead to progress. Reichenbach had declared that there was no logic of scientific method, that is, no proof or refutation. The basis for his claim that there could be no refutation was that any theory could be protected from a putative refutation with some ad hoc maneuver. Popper responded to Reichenbach with his Logik der Forschung (Logic of Research) and by introducing methodology into his deliberations. The methodological rule enabled him to avoid ad hoc protection of theories and thus enabled him to show how theories could be refuted. Third, he introduced the rule: only refutable theories—the term “fiction” no longer appears in his work—are scientific and may be deemed scientific.

This view was no longer justificationist, that is, it no longer claimed properly to assign truth values to sentences. All “assignments” are conjectural. But Popper had at that point no non-justificationist theory of rationality in general; his theory applied to science alone. He did not at that point notice problems which his theory raised for the broader framework of rationality which all philosophers of science had used since antiquity, the framework that identified the rational with the proven.

The conflict between Popper’s new theory of science and his older theory that only proven or refuted sentences can take truth values was removed by Tarski. Tarski’s definition of truth, as Tarski explained to Popper, allows for non-proven but still true sentences. Tarski thereby did away with the theory of truth that had given Popper so much trouble. Tarski did not necessarily offer Popper an adequate theory of truth for his philosophy of science. But Tarski did free him from a false theory which was a great impediment to the construction of a truly fallibilist, realist theory of science. Popper never clearly explained the importance that Tarski had for him at the time. This failure to explain how the logic of his problem changed as a result of Tarski’s theory was part of his repression of the fact that he had held a justificationist theory of truth for a long time, even after he began writing a fallibilist book. After his meeting with Tarski, he was free to develop his fallibilist theory of science in new ways, because he could claim that theories could be true even though there was no proof of them. During his earlier years in London, during 1946-1965 or so, he returned to the possibilities this fact opened up.

In Logik der Forschung Popper developed a theory of the growth of scientific knowledge without justification. But he had no general theory of rationality without justification. Indeed, he still limited rationality to science and methodology. However, at least three problems arose for this limited view of rationality.

Popper maintained at that point that scientists gain knowledge not by proofs but by refutations of good conjectures and by replacing them with new and better ones. These new conjectures avoid earlier mistakes, explain more, and invite new tests. He originally thought of this theory as eo ipso a theory of rationality: outside of science and methodology he made no allowance for rationality. He identified research, science and methodology, as the title of his book indicates.

Difficulties piled up fast. First, if rationality is limited to science, how is methodology rational? Methodology can only be rational if methodology is the empirical study of science—as Whewell said—or if non-empirical research can be rational. Popper could not view methodology as a science of science because he held that it is not merely descriptive but also prescriptive. Yet it should be rational.

The second problem arose as Popper tried to apply his methodology of the physical sciences to the social sciences. The Poverty of Historicism and The Open Society and Its Enemies defend the open society on the grounds that only open societies preserve reason, that is, criticism, and as a consequence only open societies can be civilized. But why is a choice for the open society rational? He had no answer. He merely said that the acceptance of reason was a consequence of sympathy for others. Nothing can be said to convince those to change their minds who accept the barbaric consequences of fascism or communism.

The third problem concerns metaphysics. Before he had ever developed his own philosophy of science, he had defended in his doctoral dissertation the view that metaphysical hypotheses can serve as working hypotheses in the construction of scientific theories. His discussion there merely concerned the use of physicalist metaphysics as a guide for psychological research. He said that this was fine, but one should not decide a priori that a view of psychological processes as physical is needed or even possible. Scientific research—he was not clear then what that meant—should decide this. He was later pressed, however, to decide between competing metaphysical theories with which to interpret science, even in the absence of a scientific answer. Was the world determined or not? Questions such as this raised the question as to whether one metaphysical theory can be better or worse than another and whether one could find out which one is better. He gave up his earlier view of rationality as limited to scientific research and methodology, but he still insisted that for science some metaphysical theories are merely heuristic, and no more than that.

To extend his theory that rationality consisted of scientific research and methodology alone, Popper loosened his standard of rationality. Rejecting the older standard of rationality — proof – - as too high, he began to view the standard for science, refutability, as too high for the rationality that obtains outside science. Whereas earlier he had replaced justification with refutation, he now replaced refutation with criticism. Popper thereby created a new philosophical perspective by generalizing his theory of scientific research. The name he gave to this extension is “critical rationalism.” Popper introduced it in the introduction to his Conjectures and Refutations, where he characterized it briefly as the critical attitude. He used it also to describe views he developed earlier, in The Open Society and Its Enemies.

Could his critical rationalism apply to other fields? Could various fields also not only do without (epistemological) justification but also raise their levels of rationality with the use of critical methods? Critical rationalism became a project to employ critical methods as a substitute for epistemological justification in all areas of life.

3. Joseph Agassi and William Bartley

Outside of Popper’s own efforts to develop this project, the first two most significant endeavors were undertaken by Joseph Agassi and William Bartley. Although Agassi’s efforts began somewhat earlier than Bartley’s, their development overlaps considerably; the two were in conversation with each other for much of the time that they were working out their ideas as Popper’s students. I begin with Agassi, who developed Popper’s philosophy piecemeal and then turn to Bartley who attempted to give critical rationalism a comprehensive statement, that is, a version of it which would explain how a critical rationalist could adopt a critical stance toward any idea whatsoever, including its own claims.

Agassi began with his dissertation, in which he posed the question, How can metaphysics be used to guide scientific research without making science subordinate to it? Duhem had warned that, were science to concern itself with metaphysics, it would be subordinate to it. Encouraging scientists to engage in metaphysical debates would cause dissent and lead them away from science’s main task of constructing empirical theories.

Agassi’s project was to show how metaphysical research could facilitate empirical progress without tyrannizing science. He did this by extending Popper’s theory of the methods of scientific practice to include the critical, and thereby progressive, use of metaphysical theories to guide scientific research. On his view metaphysics need not be a mere heuristic, that is, a source of ideas, but rather a systematic guide to scientific research and a provisional standard for desirable theories. Metaphysics can be useful in advancing science by giving guidelines for the search for empirical explanations and by deepening the understanding of the world offered by science. But, he also said, it can help achieve these aims only when used critically. A critical stance toward metaphysics is possible when two or more metaphysical research programs compete with each other to construct empirically refutable theories. This, he argued, is just what happened when Faraday used his metaphysical field theory as the framework within which he constructed physical (field) theories. His competitors tried to explain the same phenomena under the Newtonian assumption that all forces act at a distance. Faraday’s theory of electro-magnetic events eventually had an enormous impact, because his metaphysics enabled him to construct better physical theories than his competitors.

Bartley developed a comprehensive version of critical rationalism. He argued that there were two problems that showed Popper’s original version was too limited. Popper encountered the first of these as he wrote The Open Society and Its Enemies where he discussed the problem: Why should one be rational? He conceded that rationality is limited, as its choice is pre-rational, a decision based on feelings. Bartley viewed Popper’s problem of the limits of the ability to argue rationally in favor of rationality as parallel to a problem he (Bartley) had earlier encountered in religious philosophy: defenders of religion claim that commitment to some religion is just as rational as commitment to rationality: each individual has to choose some starting point, and each starting point must be arbitrary. Each starting point then is just as pre-rational as the other, since each choice is beyond the limits of reason.

Bartley viewed the inability to defend rationality rationally as amounting to the inability to show the superiority of rational methods to solve problems over any other method. Bartley saw this limitation as an important defect. But in Popper’s approach to rationality as critical rather than justificatory, he found a way to overcome it. For, he argued, on the one hand, the theory of rationality as proof should itself be proven, but in fact it is not provable, whereas, on the other hand, the theory of rationality as readiness to appraise theories critically should itself be open to criticism, and this is quite possible. It is then no longer the case that the adoption of a rational approach to problems is no more rational than commitments to belief systems, such as those of some religion: the theory that rational practice means holding all theories open to criticism, may itself be held open to criticism. This also means that the use of rational methods to solve problems may be rationally defended, that is, we may use rationality to answer objections to the use of rationality.

Could this theory allow one to hold religious beliefs rationally by holding them open to criticism? Bartley never answered this question explicitly. He hinted that he did believe this was the case, and some have understood him as adopting this position. Some critical rationalists are believers and some are not. Standards here remain vague. The winner of the Popper essay prize argued that Christians were also critical rationalists, because they discussed, for example, the theological significance of their religious experiences and have developed their views at Church councils. (Elliot 2004) Agassi has pointed out that the Talmudic tradition is highly critical within certain bounds, yet cannot be said to have a high degree of rationality. If all critical discussions, even those within sects, qualify their practitioners as critical rationalists, then critical rationalism itself dissolves. To take seriously the replacement of justification with criticism, Agassi suggests, requires demarcation between effective and ineffective critical methods.

Bartley called his view “comprehensively critical rationalism” to distinguish it from Popper’s critical rationalism. It should not merely explain how one can conduct rational inquires in specific fields, but it should apply to the theory of rationality itself. Bartley added a list of critical standards one may use to evaluate ideas in any area whatsoever: a proposed idea should be a solution to an important problem, internally consistent, not refuted, and consistent with science. The first three are incorporated into virtually all critical rationalist theories. The fourth has been treated with more caution: science might also be mistaken, especially when it contains competing theories. A new metaphysical alternative may be inconsistent with established physical theory, as Faraday’s was, yet be quite important for progress.

John Watkins considered Bartley’s theory a reinforced dogmatism with a “Heads I win, tails you lose” strategy: If comprehensive critical rationalism faces no effective criticism it wins, but if it does, it thereby shows that it can meet its own standards and then again it wins. This criticism overlooks the fact that, if it faces effective criticism, it is shown to be wrong. Bartley’s standard is a necessary condition of rationality, but meeting it is no reason for clinging to an effectively criticized theory.

Bartley’s ideal of holding all ideas open to criticism has been an important part of critical rationalism. But it soon became apparent that the problems of how to develop critical rationalism were more important than demonstrating just how comprehensive it could be or of maintaining this comprehensive position. In order to see how and why this realization came about it is useful to return to Agassi.

Agassi deems the focus of Bartley’s of approach to be misplaced: it unduly emphasizes the defense of rationality as rationally defensible. Rationality does not need defense; it needs improvement, Agassi says. And we may try to improve it piecemeal. We are all rational to some degree and are all interested from time to time in using reason more effectively than we now do. We cannot help but be rational, since thinking is, like seeing, innate to some extent. No one is always rational or perfectly rational any time. Our best hope, then, is to use rationality to improve the partial and limited rationality which we all use to one degree or another. We use a bootstrap process in that we use the rational methods we now have at hand to develop better methods, whereby the methods we use may very well be corrected or even discarded.

Agassi also applied Popper’s non-justificationism to the historiography of science. Like many, Popper wanted the theory of science to describe science, but he hardly tried to apply his view to the history of science. Agassi developed a far wider picture of the history of science from Popper’s viewpoint, contrasting the traditional inductivist and conventionalist historiographies with a non-justificationist one. Inductivism distorts the history of science as it is the view of innovations as either completely right or quite useless; conventionalism distorts the history of science because it explains away radical changes. John Wettersten extended this application to the historiography of psychology, explaining how a non-justificationist approach was needed to remove peculiar distortions there.

4. Critical Rationalism in Mathematics

In Proofs and Refutations Imré Lakatos extended the range of critical rationalism into mathematics. This area is just where one would expect that it would be the most difficult to develop a theory of the growth of knowledge by criticism rather than by proof, or, as Lakatos put it, by proofs and refutations. Putative counter-examples, he illustrated historically, often refute “proofs” and thus require improvements.

Lakatos did not provide for the use of frameworks to formulate problems in mathematics, nor did he discuss the rules which mathematicians should follow in formulating and criticizing proofs. He forcefully argued against premature formalization, but he did not allow for the modern method of introducing a field axiomatically from the start. His theory of response to criticism only shows that varying ways of responding to problematical cases are available.

As a beginning this is fine. But caring for the central task of critical rationalism, that is, for the development of critical methods (in mathematics) as an alternative to the quest for justifications, requires the replacement of justificationist methods with critical ones. Is this at all possible? Answers to this question might enlighten us about the rationality of mathematical research. They might supplement and/or improve Lakatos’s portrayal of mathematical research by accounts of the ways it proceeds, and explain how decisions about the direction of research are made rationally.

Several thinkers have taken up this question; but with only one exception, they have sought to use Lakatos’s justificationist methodology of scientific research programs. The exception is Peggy Marchi who broke off her research before she had constructed any developed view. Three thinkers, however, have made attempts to take Lakatos’ methodology of research programs in a critical spirit and then apply it to the history of mathematics.

D.D. Spalt (Spalt 1981) argues that Lakatos’ methodology of research programs is inapplicable to the history of mathematics as mathematicians are more open skeptical and critical than Lakatos’ methodology describes. This confirms Lakatos’s turn away from a critical approach, but does not help us further since it does not go on to ask if a genuinely critical approach, say, to the use of research programs such as Agassi’s would help us. But Spalt also finds no mathematicians who follow any clear research program at all. He defends a view of mathematics which has great similarity to Feyerabend’s view of science: there is no methodology which can describe all mathematical research.

G. Giorello (Giorello 1981) argues that Lakatos’ theory of scientific research programs is better applicable to mathematical research than Popper’s or Kuhn’s. Teun Koetsier in turn found Giorello’s argument inadequate (Koetsier 1991 pp. 145ff.). This is not surprising, since Lakatos’ methodology of research programs is sketchy.

Koetsier was not satisfied with Giorello’s vague results nor with Spalt’s negative ones. He proposed a revision of Lakatos’ theory which would enable him to describe how mathematical research proceeds. His revised version is closer to Agassi’s theory of research programs, which, Agassi suggested, might be used to explain how mathematical problems were chosen and how mathematical research was coordinated.

Lakatos’s historical reconstructions of mathematical developments are Popperian in that they portray not only mathematical theorems and their proofs, but also their refutations, and their replacement by new ones. Koetsier criticizes this portrayal. He finds instead that the aim of mathematical research has been directed at refining mathematical theorems. The refined theorems are then by and large accepted and entered into the body of mathematical knowledge, where they then stay, subject only to further refinements. Koetsier agrees, however, that Lakatos’s theory does show how mathematicians work when solving problems within some narrowly defined areas of research. This research is fallible, he agrees, and this allows it to progress by the discovery of difficulties with previous theories which are overcome by succeeding ones.

Koetsier discusses clusters of mathematical theories that are part of identifiable research traditions. These traditions pose their own problems and are identifiable by their offerings of clusters of mathematical theorems. Each tradition, however, is not replaced by some competing one as in the case of science, where one explanation is superseded by another leading to the rejection of the former. In this respect the theories of science of Popper and/or Lakatos cannot be applied to the history of mathematics. Rather, each theory progresses in its domain and the results it produces are largely cumulative.

In order to explain how progress is made in such research traditions, Koetsier employs the suggestion made by Marchi that theorems should be taken as analogous to facts. (Marchi 1976) Whereas scientists seek to explain facts, mathematicians seek to prove theorems. Theorems are, just as facts, accepted provisionally. Instead of seeking to explain them as in science, mathematicians seek to prove them. Mathematics grows as new theorems are discovered and proved.

This theory leads back to the problem posed by Agassi and Marchi: How is research coordinated? Koetsier finds that Lakatos’s theory of mathematics describes “local” mathematical research rather well. It describes how they solve problems within some cluster of theories and/or methods. He finds various research traditions, which have been used to set problems. But, he does not explain how such traditions arise nor why they are chosen. Mathematical research is, then, coordinated by interests in particular kinds of mathematical objects and/or particular methods. But, how are these chosen? Why do they change?

Koetsier also faces the question: Which theorems should mathematicians prove and why? He notes that some are central and others that seem simply too ad hoc to bother with. But, how does one decide? He offers a list of measures by which to judge the importance of theorems. His list of methods for appraising the ad hoc nature of theorems is interesting but still rather ad hoc. (Koetsier p. 170-171)

Agassi’s theory of metaphysical research programs might have helped him here. Unlike Lakatos’ inferior and subsequent theory, Agassi’s was designed to solve the problems of “How is scientific research coordinated?” or “How do scientists choose their problems?” and “How can we explain simultaneous discoveries?” His answer is that problems in science are often chosen for their relevance to metaphysical problems. He developed at some length and in some depth the conflict between Faraday’s field theory and Newton’s atomic theory to show how problems were chosen which bore on this controversy and how the two metaphysical research programs could compete against each other.

How much of the choice of problems in the history of mathematics can be explained by Agassi’s conjecture that they are regularly chosen due to their relevance to metaphysical problems? This is still an open question. But some problems clearly were. Among them are problems concerning irrational numbers, whether numbers exist in a Platonic world, problems concerning the nature of infinitesimals or irrational numbers or the square root of minus one or the nature of transfinite numbers as well as questions concerning the possibilities of non-Euclidean geometries. A history of mathematics written from this point of view might be enlightening, if it could portray underlying metaphysical concerns as focusing mathematical research on certain kinds of problems and the development of methods to deal with them.

It should be noted that J. O. Wisdom had

المصدر: Internet Encyclopedia of Philosophy

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